Q: Which of the following is the correct formula for a two-sample t-interval for the difference in means assuming unequal variances?

$(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2, \nu} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

Question 1

A researcher wants to test whether there is a relationship between gender and favorite sport using a chi-square test of independence. Which of the following is the correct null hypothesis for this test?

Accepted Answer

H_0: Gender and favorite sport are independent.

Question 2

Given the following observed and expected counts for a cell in a contingency table: observed = 40, expected = 25, what is the contribution of this cell to the chi-square statistic?

Accepted Answer

$\frac{(40-25)^2}{25} = 9$

Question 3

A chi-square test of independence yields a p-value of 0.0453. At the 5% significance level, what is the correct conclusion?

Accepted Answer

There is a significant relationship between the two variables at the 5% significance level.

Question 4

Which of the following statements about the chi-squared distribution is FALSE?

Accepted Answer

The chi-squared distribution is symmetrical.

Question 5

A 95% confidence interval for the difference in means between two machines is (0.0155, 0.424). What is the correct interpretation of this interval?

Accepted Answer

We are 95% confident that the true difference in mean weights produced by the two machines is between 0.0155 and 0.424.

Question 6

In a hypothesis test comparing two means, $H_0: \mu_1 = \mu_2$, the test statistic is calculated as $t = 2.7488$ with a critical value of $t_{\alpha} = 2.0$. What is the correct conclusion at the $\alpha = 0.05$ significance level?

Accepted Answer

Reject $H_0$; there is sufficient evidence to conclude the means are different.

Question 7

A matched pairs design is most appropriate when:

Accepted Answer

Each subject is measured under both conditions, and the differences are analyzed.

Question 8

Which of the following is the correct formula for a two-sample t-interval for the difference in means assuming unequal variances?

Accepted Answer

$(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2, 
u} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

Question 9

A p-value is best described as:

Accepted Answer

The probability of observing a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true.

Question 10

Which of the following is a correct interpretation of a 99% confidence interval for a population mean?

Accepted Answer

If we were to take many samples and build a confidence interval from each, about 99% of those intervals would contain the true population mean.

Question 11

A one-way ANOVA is performed using four populations, each with $\sigma^2 = 15$ and equal sample sizes. What is the expected value of the mean square error (MSE)?

Accepted Answer

225

Question 12

Which of the following is NOT an assumption of the equal variances (pooled) t-test?

Accepted Answer

The two samples must have equal means.

Question 13

In a study, the mean cost to repair cars with ABS is $145.42536 and without ABS is $360.48. The test statistic for the difference in means is $-2.4788$. What is the correct conclusion at the 5% significance level?

Accepted Answer

At the 5% significance level, we can say that the mean cost to repair cars with ABS is lower than the mean cost to repair cars without ABS.

Question 14

Which of the following best describes a matched pairs design?

Accepted Answer

Each subject receives both treatments, and the difference is analyzed.

Question 15

Which test is most appropriate to determine if the mean time spent cleaning up in unsuccessful companies is greater than in successful companies, given unequal variances and large sample sizes?

Accepted Answer

2-sample t-test for means assuming unequal population variances

Question 16

A survey asks respondents to select their favorite breakfast cereal from a list of five options. What type of variable is 'favorite cereal'?

Accepted Answer

Categorical (nominal)

Question 17

Which of the following is the correct formula for the standardized residual in a chi-square test?

Accepted Answer

$\frac{O - E}{\sqrt{E}}$

Question 18

In a two-sample z-test for proportions, which of the following is the correct test statistic formula?

Accepted Answer

$z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$

Question 19

A study is conducted to determine if there is a difference in the mean commute time between students who carpool and those who drive alone. Which statistical test is most appropriate?

Accepted Answer

Two-sample t-test for independent means

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